$f(x, y) = (3x^2y, 3x)$ What is the curl of $f$ at $(-3, 1)$ ?
The formula for curl in two dimensions is $\text{curl}(f) = \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}$, where $P$ is the $x$ -component of $f$ and $Q$ is the $y$ -component. Let's differentiate! $\begin{aligned} \dfrac{\partial Q}{\partial x} &= \dfrac{\partial}{\partial x} \left[ 3x \right] \\ \\ &= 3 \\ \\ \dfrac{\partial P}{\partial y} &= \dfrac{\partial}{\partial y} \left[ 3x^2y \right] \\ \\ &= 3x^2 \end{aligned}$ Therefore: $\text{curl}(f) = 3 - 3x^2$ The curl of $f$ at $(-3, 1)$ is $-24$.